# Coordinate Rotation

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## Main Question or Discussion Point

In Mathematical Methods for Physicists, 6th Edition, by Arfken and Weber, Chapter 1 Vector Analysis, pages 8-9, the authors make the following statement:

"If Ax and Ay transform in the same way as x and y, the components of the general two-dimensional coordinate vector r, they are the components of a vector A. If Ax and Ay do not show this form invariance (also called covariance) when the coordinates are rotated, they do not form a vector."

I understand how to use equations (1.9) and their derivations, but could anyone please explain the above statement?

Thank you so much for your help...

Stephen Tashi
In Mathematical Methods for Physicists, 6th Edition, by Arfken and Weber,
For those interested, this edition of the book is available online as a PDF - perhaps legally?

Chapter 1 Vector Analysis, pages 8-9, the authors make the following statement:

"If Ax and Ay transform in the same way as x and y, the components of the general two-dimensional coordinate vector r, they are the components of a vector A. If Ax and Ay do not show this form invariance (also called covariance) when the coordinates are rotated, they do not form a vector."
I find that passage to be confusing. Here's my guess at what it means: Suppose some physical phenomenon is assigned cartesian coordinates $(A_x,A_y)$ and has coordinates $(A'_x, A'_y)$ in a rotated coordinate system. Is the physical phenomenon a vector? The passage says that if $(A'_x, A'_y)$ can be computed from $(A_x,A_y)$ in the same way we would compute the new coordinates for a geometric point $(A_x,A_y)$ in a rotated coordinate system then the phenomenon is a vector.

For that passage to have significance, you must be able to imagine that there physical phenomenon described by two numbers $(A_x,A_y)$ whose coordinates in a rotated coordinate system cannot be computed by imagining $(A_x,A_y)$ to be the cartesian coordinates of a point and computing the new coordinates as we would compute the new coordinates for a geometric point.

Examples the authors give for such phenomena are "elastic constants" and "the index of refraction in isotropic crystals". Perhaps experts on those topics can elaborate.

If there is a phenomenon with a "magnitude and direction", it is tempting to think that it must be a vector and that it can be represented as an arrow from the origin of a cartesian coordinate system to some point in the coordinate system. The book says such a representation doesn't work for some phenomena.

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